Here \(\sigma =e(n_i-n_e)\)  is the net charge density, \(\mathbf {J}\)  is the current, and the speed of light is related to the permeability and permittivity of vacuum with \(c=(\mu _0 \varepsilon _0)^{-1/2}\) .
To ensure the divergence constraints in numerical simulations, some techniques can be used in solving Maxwell's equations.
Here we employ the perfectly hyperbolic Maxwell's equations (PHM) [1]},
\(\begin{aligned}&\frac{\partial \mathbf {E}}{\partial t} - c^2 \nabla _\mathbf {x} \times \mathbf {B} + \chi c^2 \nabla _\mathbf {x} \phi = - \frac{1}{\varepsilon _0} \mathbf {J}, \\&\frac{\partial \mathbf {B}}{\partial t} + \nabla _\mathbf {x} \times \mathbf {E} + \gamma \nabla _\mathbf {x} \psi = 0, \\&\frac{1}{\chi } \frac{\partial \phi }{\partial t} + \nabla _\mathbf {x} \cdot \mathbf {E} = \frac{\sigma }{\varepsilon _0}, \\&\frac{\varepsilon _0 \mu _0}{\gamma } \frac{\partial \psi }{\partial t} + \nabla _\mathbf {x} \cdot \mathbf {B} = 0, \\\end{aligned}\) 
